Dynamic resource allocation using projected future benefits

ABSTRACT

A method for server allocation in a Web server “farm” is based on limited information regarding future loads to achieve close to the greatest possible revenue based on the assumption that revenue is proportional to the utilization of servers and differentiated by customer class. The method of server allocation uses an approach of “discounting the future”. Specifically, when the policy faces the choice between a guaranteed benefit immediately and a potential benefit in the future, the decision is made by comparing the guaranteed benefit value with a discounted value of the potential future benefit. This discount factor is exponential in the number of time units that it would take a potential benefit to be materialized. The future benefits are discounted because by the time a benefit will be materialized, things might change and the algorithm might decide to make another choice for a potential (even greater) benefit.

CROSS REFERENCE TO RELATED APPLICATION

The subject matter of this application is related to application Ser.No. 10/000,320 filed concurrently herewith by Tracy Kimbrel et al. For“Dynamic Resource Allocation Using Known Future Benefits”. The subjectmatter of application Ser. No. 10/000,320 is incorporated herein byreference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to benefit task systems and,more particularly, to a policy for allocating resources to maximize somebenefit. The invention may be applied to a variety of problems, and thebenefit may be either tangible (e.g., profit) or intangible (e.g.,customer satisfaction). In a specific example, the invention hasparticular application to server allocation in a Web site server “farm”given limited information regarding future loads to maximize profits forthe Web hosting service provider. In another specific example, theinvention can be applied to the allocation of telephone help in a way toimprove customer satisfaction. In yet another example, the invention maybe applied to distributed computing problems where the resources to beallocated are general purpose computers connected in a network and usedto solve computationally intensive problems.

2. Background Description

Web content hosting is an important emerging market. Data centers andWeb server “farms” are proliferating. The rationale for using suchcenters is that service providers can benefit from economies of scaleand sharing of resources among multiple customers. This benefit in turntranslates to lower cost of maintenance for the customers who purchasethese hosting services. Web content hosting services are structured inmany ways. One of the most prevailing ways is outsourcing: the customersdeliver their Web site content in response to HTTP (hyper text transferprotocol) requests. Service providers will use “farms” of commodityservers to achieve this goal.

One of the components in the payment for such a service is “pay perserved request”. Thus, one of the main objectives of the serviceprovider is to maximize the revenue from served requests while keepingthe tab on the amount of resources used. Ideally, the allocation to aWeb site should always suffice to serve its requests. However, due to alimited number of servers and the overhead incurred in changing theallocation of a server from one site to another, the system may becomeoverloaded, and requests may be left unserved. Under the assumption thatrequests are not queued, a request is lost if it is not served at thetime it is requested. The problem faced by the Web hosting serviceprovider is how to utilize the available servers in the most profitableway. What adds to the complexity of this problem is the fact that futureload of the sites is either unknown or known only for the very nearfuture. A Web content hosting service provider therefore needs aprocedure to dynamically allocate servers in a server “farm” to itscustomers' Web sites.

Similar considerations apply in the cases of computer servers andtelephone support centers. Telephone support centers typically arecomputer controlled telephone networks having a number of technicalsupport, order support and customer service support operators. Theseoperators are resources that must be allocated to customers who call in.Computer software is used to answer telephone calls and direct the callsto the appropriate pool of operators. In this application, the operatorsare the resources to be allocated. The wait time that a customerexperiences is inversely proportional to customer satisfaction and,therefore, it is important to be able to dynamically allocate resourcesin such a manner as to minimize customer wait time and increase customersatisfaction. In this application, customer benefit is the intangiblebenefit which is sought to be maximized.

In yet another example, the resources to be allocated are generalpurpose computers used to solve computationally intensive problems. Inthis environment, multiple computers can be used concurrently to solve aproblem faster than a single computer can solve it. The computers wouldbe connected in a network which may include the Internet. It has evenbeen proposed that personal computers connected to the Internet mightconstitute resources that could be employed in solving such problems. Itis anticipated that a market for such services will become standardizedto some extent, so that the computer cycles become a commodity(resource) available from multiple vendors. It would be an advantage todynamically allocate such resources.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide a methodfor server allocation given limited information regarding futurerequirements.

It is another object of the invention to provide a method for serverallocation to achieve close to the greatest possible benefit based onthe assumption that benefit is proportional to the utilization ofresources.

According to the invention, the method of resource allocation uses anapproach of “discounting the future”. Specifically, when the policyfaces the choice between a guaranteed benefit immediately and apotential benefit in the future, the decision is made by comparing theguaranteed benefit value with a discounted value of the potential futurebenefit. This discount factor is exponential in the number of time unitsthat it would take a potential benefit to be materialized. The futurebenefits are discounted because by the time a benefit will bematerialized, things might change and the algorithm might decide to makeanother choice for a potential (even greater) benefit.

In the practice of the invention, the resource allocation problem ismodeled mathematically. In the model, time is divided into intervals.For the Web server farm problem, the assumption is made that each site'sdemand is uniformly spread throughout each such interval. Serverallocations remain fixed for the duration of an interval. It is alsoassumed that servers are reallocated only at the beginning of aninterval, and that a reallocated server is unavailable for the length ofthe interval during which it is reallocated. This represents the time to“scrub” the old site (customer data) to which the server was allocated,to reboot the server and to load the new site to which the server hasbeen allocated. The length of the time interval is set to be equal tothe non-negligible amount of time required for a server to prepare toserve a new customer. In current technology, this time is in the orderof 5 or 10 minutes.

Each server has a rate of requests it can serve in a time interval. Forsimplicity, all rates are assumed to be identical. Due to practicalconcerns (mainly security constraints placed by customers), sharing ofservers at the same time is not allowed. That is, customers shareservers only in the sense of using the same servers at different times,but do not use the same servers at the same time. Thus, even in case ofoverload, some of the servers may be underutilized if they are allocatedto sites with rates of requests lower than the servers' rate.

Each customer's demand is assumed to be associated with a benefit gainedby the service provider in case a unit demand is satisfied. Given afixed number of servers, the objective of the service provider is tofind a time-varying server allocation that would maximize benefit gainedby satisfying sites' demand. In a fully online implementation of theinvention, only the demand of the current interval is known at thebeginning of the interval. It is assumed that some amount of futuredemand is known to the service provider, and this is modeled bylookahead, which means that the demand of the sites is known for thenext predetermined number of time intervals (following the current timeinterval). Lookahead requires that some forecasting mechanism is used topredict future demands. In the online setting, an online allocationalgorithm is measured by its competitive ratio; that is, the maximumover all instances of the ratio of the benefit gained by the optimaloffline allocation to the benefit gained by the online allocation on thesame instance. Almost optimal algorithms (both deterministic andrandomized) are used for any positive lookahead. The quality of thesolution improves as the lookahead increases.

Interestingly, the model can be cast as a more general benefit tasksystem. In this task system, we are given a set of states for each time,t, and a benefit function. The system can be at a single state at eachtime, and the benefit for time t is a function of the system states attimes t−1 and t. The goal is to find a time varying sequence of statesthat maximizes the total benefit gained. That is, at each time t, weneed to determine to which state should the system move (and this willbe the state of the system at time t+1), and we gain the benefit that isdetermined by the benefit function. Similar to the server farm model, inthe fully online implementation of the invention, only the benefit ofthe current step is known at the time the next move is determined, andin case of lookahead, L, the benefits of the next L steps (in additionto the current) are known.

It can be shown that benefit task systems capture also benefitmaximization variants of well studied problems, such as the k-serverproblem (see A. Borodin and R. El-Yaniv in On-Line Computation andCompetitive Analysis, Cambridge University Press, 1998) and metricaltask systems (see On-Line computation and Competitive Analysis, citedabove). Thus, our results hold for these variants as well, and show thatthe benefit variants of these problems may be more tractable than theircost minimization variants.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, aspects and advantages will be betterunderstood from the following detailed description of a preferredembodiment of the invention with reference to the drawings, in which:

FIG. 1 is a block diagram illustrating the architecture of a Web serverfarm;

FIG. 2 is a flow diagram illustrating the process of allocating serversusing projected future benefits; and

FIG. 3 is a flow diagram illustrating the process of computing newallocations of servers.

DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT OF THE INVENTION

Although the invention is described in terms of a specific applicationto a Web server farm, this explanation is by way of example only. Itwill be understood by those skilled in the art that the invention may beapplied to other applications. Among those applications are the customertelephone support problem and the allocation of computers tocomputationally intensive problems already mentioned. The Web serverfarm problem will serve to provide a concrete application of theinvention which can be applied to other resource allocation problems.

Referring now to the drawings, and more particularly to FIG. 1, there isshown, in generalized form, the architecture of a Web server farm of thetype managed and maintained by a Web hosting service provider. The farmitself comprises a plurality of servers 10 ₁, 10 ₂, 10 ₃, . . . , 10_(n) connected to a server farm network 11. The server farm network 11connects the servers via the Internet 12 to a plurality of Web browsers13 ₁, 13 ₂, . . ., 13 _(n). Customers of the Web hosting serviceprovider purchase hosting services, which include maintaining thecustomers' Web sites so that the Web sites can be readily accessed bypersons using the Web browsers. The Web hosting service provider seeksto maximize profits by allocation of resources, i.e., the servers. Thisis the function of the dynamic resource allocator 14 connected to theserver farm network.

FIG. 2 illustrates the general process implemented by the dynamicresource allocator 14. The process begins in function block 21 where thecurrent and forecasted per-customer demands and revenues are obtained.Next, current allocations of servers are obtained in function block 22,and then new allocations are computed in function block 23. The computednew allocations are compared with the current allocations in decisionblock 24 and, if they are different, then in function block 25,re-allocated servers are directed to serve their new customers beforethe process loops back to function block 21. If the new allocations arenot different from the current allocations, as determined in decisionblock 24, the process goes directly to function block 21 to begin theprocess anew.

The first function block 21, obtain current and forecasted per-customerdemands and revenues, is outside the scope of the present invention. Weassume some forecasting mechanism is used to determine the projecteddemands and benefits. As will be explained in more detail, the presentinvention is the algorithm for deciding what the allocations should bebased on a given forecast; i.e., function block 23, compute newallocations. Allocations are made based on the result of thiscomputation.

The Web Server Farm Problem

Suppose that we are given s Web sites that are to be served by k Webservers. (For simplicity, we assume that all servers are identical.)Time is divided into units. It is assumed that the demand of a Web siteis uniform in each time unit. Each server has a “service rate” which isthe number of requests to a Web site each server can serve in a timeunit. Without loss of generality, we normalize the demands by theservice rate so that a server can serve one request per time unit anddemands of a site may be fractional. A Web server can be allocated to nomore than one site at each time unit and it takes a time unit to changethe allocation of a server.

A problem instance consists of the number of servers, k, the number ofsites, s, a non-negative benefit matrix, b_(i,t), denoting the benefitgained by serving a request of site i∈[1 . . . s] for time step t≧1, anda non-negative demand matrix, {d_(i,t)}, denoting the number of requestsat site i for time step t. The goal is to find for each site i∈[1 . . .s] a time varying allocation {a_(i,t)} of servers, so as to maximize thetotal benefit, as follows. The allocation must satisfy that for each t,

${{\sum\limits_{i = 1}^{s}a_{i,t}} \leq {{k.\mspace{14mu}{Only}}\mspace{14mu} a_{i,t}^{\prime}}} = {\min\left\{ {a_{i,{t - 1}},a_{i,t}} \right\}}$of the servers allocated to site i for the time step t are “productive”,i.e., actually serve requests. We get that the total benefit of anallocation {a_(i,t)} is

${\sum\limits_{t \geq 1}{\sum\limits_{i = 1}^{s}{{b_{i,t} \cdot \min}\left\{ {d_{i,t},a_{i,t}^{\prime}} \right\}}}} + {\sum\limits_{t \geq 1}{\sum\limits_{i = 1}^{s}{{b_{i,t} \cdot \min}{\left\{ {d_{i,t},a_{i,t},a_{i,{t - 1}}} \right\}.}}}}$

In the online solution of the problem according to the presentinvention, we need to compute the allocation {a_(i,t)} at time t giventhe demands so far, i.e., {d_(i,t′)} for all i and t′≦t. It is notdifficult to show that any fully online algorithm may performarbitrarily badly with respect to the offline optimal solution. This istrue because of the one time unit lag in a server's availability. Thus,we consider online algorithms with lookahead. In the online verison withlookahead L, at each time t we are additionally given the demands fortimes t+1, . . . , t+L, i.e., the entries d_(i,t′) for i ∈[1 . . . s]and t+1≦t′≦t+L, and we need to compute the allocation at time t, i.e.,a_(i,t) for i∈[1 . . . s].

The Benefit Task System Problem

The Web server farm problem is a special case of the generalized tasksystem benefit problem. In this problem, we are given (i) a set ofpossible states U_(t), for each time t≧0 and (ii) a non-negative benefitfunction B whose domain is ∪_(t)(U_(t)×U_(t+1)), that denotes, for eachtime t>0, the benefit that is accrued (at time t+1) by the transitionfrom a state U_(t) to a state U_(t+1). The goal is to choose a state stfor each time t so as to maximize the total benefit

$\sum\limits_{t}{{B\left( {s_{t},s_{t + 1}} \right)}.}$

In the online version of the problem with lookahead L, the states_(t)∈U_(t) should be computed based on knowing only U_(t′) for t′≦t+L,and the restriction of the function B to pairs of these sets of states.

Observe that the Web server farm problem can be cast in this setting byidentifying each possible allocation of servers to sites at time t witha state S_(i,t), and defining the benefit function B(S_(i,t),S_(j, t+1)) to be the benefit gained by changing the allocation at timet from the one represented by S_(i,t) to the allocation at time t+1represented by S_(j, t+)1, (In a sense, the set of states for all timesis the same.) The number of states is exponential in the number ofservers k, so the states and the benefit functions are implicit andfollow from the more succinct representation of the Web server farmproblem. For example, the values B(s_(i, t), s_(j, t+1)) are not listedexplicitly, and any single value can be efficiently computed whennecessary.

Benefit maximization variants of well known problems can also be cast inthis setting of a benefit task system. Consider the benefit version ofthe k-server problem with one difference. Instead of minimizing the costof satisfying all requests, we define for each time t and any possiblepair of server configurations, one at time t−1 and one at time t, a netbenefit gained by satisfying the request starting from the configurationat time t−1 and ending at the configuration at time t. (Note thatconfiguration at time t must include at least one server at the point ofthe request.) This benefit has to be non-negative and is composed of afixed positive component that is reduced by the cost to move from theconfiguration at time t−1 to the configuration at time t. The goal inthis case is to maximize the total benefit. It is not difficult to seethat this problem can also be modeled by the benefit task system definedabove and thus all our results apply to the benefit version of thek-server problem.

Similarly, consider the benefit version of a metrical task system. Thisversion is similar to the classical metrical task system (see, forexample, A. Borodin, N. Linial and M. E. Saks, “An optimal on-linealgorithm for metrical task system”, J. ACM, 39 (1992), pp. 745-763),with the difference that each task is associated with a vector ofbenefits, one for each state, such that the net benefit aftersubtracting the transition cost is non-negative. This model as well canbe cast as a benefit task system and our results apply.

Deterministic Online Algorithms for Benefit Task Systems

Here, we describe two deterministic algorithms for benefit task systemswith lookahead L. The first algorithm is a future discounting algorithmthat achieves a competitive ratio of

$\left( {1 + \frac{1}{L}} \right){\sqrt[L]{L + 1}.}$Asymptotically in L, this ratio is

$1 + {\frac{\Theta\left( {\log\; L} \right)}{L}.}$The second algorithm is an intermittent reset algorithm that achieves acompetitive ratio of

${1 + \frac{4}{\left( {L - 7} \right)}},$which is better than the first algorithm when L is sufficiently large.The Future Discounting Algorithm

This is a deterministic algorithm for the benefit task system problemthat achieves competitive ratio

$\left( {1 + \frac{1}{L}} \right){\sqrt[L]{L + 1}.}$The algorithm is based on discounting future benefits in an exponentialway. Consider a strategy that collects L+1 benefits b₀, b₁, . . . ,b_(L) (in this order) in the next L+1 time steps, we define itsanticipated benefit to be b₀+b₁/α+ . . . +b_(L)/α_(L) (for a parameterα>1 that we later choose as α=^(L)√{square root over (L+1)}).

The discounting algorithm greedily follows at each time step thestrategy with the maximum anticipated benefit. That is, at every timestep the algorithm finds a strategy for the next L+1 time steps whoseanticipated benefit is maximal and acts (in this time step) according tothis strategy. We stress that such a calculation is made every timestep, so the strategy chosen in the current time step does not have toagree with the one chosen in the previous time step.

We note that this algorithm requires that a strategy with the greatestanticipated benefit can be computed efficiently. The following lemmaprovides this requirement when the input describes the statesexplicitly. Note that we cannot apply this lemma to the Web server farmproblem since here the states are implicit and of exponential size, sowe use the offline algorithm described in copending patent applicationSer. No. 10/000,320.

Lemma 1. A strategy with the largest anticipated benefit can be computedin time that is polynomial in the number of states.

Proof. Use dynamic programming that goes interactively over the timesteps and accumulate discounted benefits. As the additional benefit ofthe next time step depends only on the current state, it suffices tohave a table of size proportional to the number of states.Theorem 2. The above discounting algorithm has competitive ratio at most

$\left( {1 + \frac{1}{L}} \right){\sqrt[L]{L + 1}.}$Proof. Consider the online algorithm at time t, and denote by b₀, b₁, .. . ,b_(L) the sequence of benefits in the strategy with the largestanticipated benefit over all strategies that the online algorithm has attime t. Define ON_(t)=b₀ and ON_(t+1) ^(≠L)=b₁/α+ . . . + b_(L)/α^(L),then ON_(t)+ON_(t+1) ^(≠L) is the largest anticipated benefit over allthe strategies that are available to the online algorithm at time t.Since the online algorithm follows this strategy at time t, ON_(t) isthe benefit that the online collects for this time step.

Fixing an offline algorithm arbitrarily, let OFF_(t) denote the benefitthat this offline algorithm collects at time t, and let OFF_(t) ^(l) bea shorthand for OFF_(t)+1/α+ . . . +OFF_(t+L)/α^(L).

One strategy that is considered by the online algorithm at time t is tofirst join the offline algorithm (i.e., move to the location of theoffline at time t+1), and then follow the offline algorithm for the nextL steps. The anticipated benefit of this strategy is at least OFF_(t)^(L). Since the online algorithm follows at each time step the strategywith the maximum anticipated benefit, we get thatON _(t) +ON _(t+1) ^(≠L) ≧OFF _(t+1) ^(L)  (1)

Another strategy that is considered by the online algorithm at time t isto follow the strategy that had the largest anticipated benefit at theprevious time t−1. (Since the first step of that strategy has alreadytaken at time t−1, we need to add an arbitrary move at the end.) Thecontribution of these benefits to the anticipated benefit at time t islarger by a factor of α than their contribution at time t−1, and so theanticipated benefit of this strategy is at least αON_(t) ^(≠L). Sincethe online algorithm follows at each time step the strategy with themaximum anticipated benefit, we get thatON _(t) +ON _(t+1) ^(≠L) ≧αON _(t) ^(≠L)  (2)Adding

$\left( {1 - \frac{1}{\alpha}} \right)$times inequality (1) and

$\frac{1}{\alpha}$times inequality (2), we get that

$\begin{matrix}{{{ON}_{t} + {ON}_{t + 1}^{\neq L}} \geq {{ON}_{t}^{\neq L} + {\left( {1 - \frac{1}{\alpha}} \right){OFF}_{t + 1}^{L}}}} & (3)\end{matrix}$Adding up inequality (3) over all time steps t (and assuming that thedemands table starts and ends with L time units of zero demand), we getthat

${{\sum\limits_{t}{ON}_{t}} \geq {\left( {1 - \frac{1}{\alpha}} \right){\sum\limits_{t}{OFF}_{t + 1}^{L}}}} = {\left( {1 - \frac{1}{\alpha}} \right)\left( {\frac{1}{\alpha} + \mspace{11mu}\ldots\mspace{11mu} + \frac{1}{\alpha^{L}}} \right){\sum\limits_{i}{{OFF}_{t}.}}}$The last equality follows since every benefit that the offline collectsis accumulated with each of the discounts 1/α, . . . , 1/α^(L). Weconclude that the competitive ratio of the algorithm is (choosingα=^(L)√{square root over (L+1)}):

${\frac{\sum\limits_{i}{OFF}_{t}}{\sum\limits_{t}{ON}_{t}} \leq \frac{\frac{\alpha}{\alpha - 1}}{\frac{\alpha^{L} - 1}{\alpha^{L}\left( {\alpha - 1} \right)}}} = {\frac{\alpha^{L + 1}}{\alpha^{L - 1}} = {\frac{\left( {L + 1} \right)^{\frac{({L + 1})}{L}}}{L} = {\left( {1 + \frac{1}{L}} \right){\sqrt[L]{L + 1}.}}}}$An Intermittent Reset Algorithm

This is a deterministic intermittent reset online algorithm that hascompetitive ratio

$1 + {O\left( \frac{1}{L} \right)}$for the benefit task system problem. This algorithm is motivated by therandomized algorithm described below that wastes one of every L+1 stepsin order to gain the optimal benefit between wasted steps. Here, wewaste one in every Θ(L) steps, but the choice of steps to waste is donecarefully in a deterministic fashion and the overall benefit that wegain is close to that of the randomized algorithm.

The algorithm works in iterations each of length at least L/2. Let s_(i)denote the start time of iteration i. Each iteration i consists of foursteps as follows:

-   STEP 1: Consider all times t=s_(i)+L/2+1 . . . , s_(i)+L. For each    such time t, let x_(t) denote the maximum benefit of any transition    from a state at time t to a state at time t+1. Let T be the time t    that minimizes x_(t), i.e., x_(T)=min{x_(t):s_(t)+L/2+1≦t≦s_(i)+L}.-   STEP 2: Compute the optimal sequence of transitions that starts at    any state at time s_(i)+1 and ends at any state at time T. This can    be done either by dynamic programming similar to Lemma 2 or by    applying the offline algorithm above. (We remark that in the first    iteration, the initial state is given and thus we may compute the    optimal sequence of transitions from the initial state to any state    at time T. The next step has to be modified accordingly.)-   STEP 3: Move from the current state at time s_(i) to the state at    time s_(i)+1 that is the starting state of the optimal sequence    computed above, and continue with the optimal sequence of moves to    the state at time T.    STEP 4: Set the starting point s_(i+1)=T for the next iteration.    Theorem 3. The above intermittent reset algorithm has competitive    ratio at most

$1 + {\frac{4}{L - 7}.}$Proof We assume L is a multiple of 4, and otherwise round down to thenearest multiple of 4 (in the expense of-increasing the competitiveratio as explained later). We will call the starting times of theiterations “gaps”. Denote the gaps by s₁, S₂, . . . We divide thetransitions taken by the algorithm into two components, the moves takenin the gaps and the steps taken in the rest of the times. Consider thepath taken by the online algorithm. It may not collect any benefit inthe gaps, but since it computes the optimal paths between gaps (allowingarbitrary initial and terminal states adjacent to the gaps), clearly thebenefit of the optimal offline between gaps is bounded by the benefit ofthe online algorithm between gaps.

Now we bound the benefit of the optimal algorithm in the gaps. Consideriteration i starting at si and computing T=s_(i)+1. Consider the portionof the online path from s_(i)+L/2+1 to si+1. Denote the length of thisportion by a, i.e., a=s_(i+1)−(s_(i)+L/2+1). Consider the portion of thepath computed in the next iteration from s_(i+1)+1 to s_(i)+L+1. Denotethe length of this portion by b, i.e., b=s_(i)+L+1−(s_(i+1)+1). By ourchoice of L, L/2 is even and a+b=L/2−1. Thus, one of a and b is even andthe other is odd. Assume a is odd and b is even; the proof is similar inthe opposite case. (We note that ideally, we would like both a and b tobe even, but choosing a+b to be even may lead to a worse case when bothare odd.)

We claim that the benefit achieved by the online algorithm in theseportions of the path at least x_(T)·(a+b−1)/2. This is true since inevery time step in the range there is a transition of benefit x_(T) andthus the online algorithm can achieve benefit at least x_(T) on everyother transition, even if it collects no benefit on the other half ofthe transitions. Since a is odd, it can collect at least x_(T)·(a−1)/2in the first portion, and since b is even, it can collect at leastx_(T)·b2 in the second.

Putting these together along with a+b=L/2−1, we have that the onlinebenefit is at least

${x_{T} \cdot \left( {\frac{a - 1}{2} + \frac{b}{2}} \right)} = {x_{T} \cdot {\frac{L - 4}{4}.}}$Since s_(i)+L+1 is always less than s_(i+1)+L/2+1, these portions aredisjoint among iterations. Summing over all iterations, we have

${{{ON} \cdot \left( \frac{4}{L - 4} \right)} \geq {\sum\limits_{i}x_{T_{1}}}},$where ON denotes the total benefit collected by the online algorithm andx_(T) _(i) denotes the maximum benefit possible in the i'th gap.Accounting for the portions outside the gaps, we have

${{ON} \geq {{OFF} - {\sum\limits_{i}x_{T_{i}}}}},$where OFF denotes the optimal offline benefit. Putting these togetherand allowing for values of L that are not multiples of 4 yields thedesired bound of

$1 + {\frac{4}{L - 7}.}$

Special Cases of the Server Allocation Problem

We devise a deterministic online algorithm for the Web server farmproblem with one server, two sites and lookahead L=1.

One server and Two Sites with Lookahead One

We devise a deterministic online algorithm for the Web server farmproblem with one server, two sites and lookahead L=1. The algorithm hascompetitive ratio T=1+φ, where

$\phi = {\frac{1 + \sqrt{5}}{2} \approx 1.618}$is the golden ratio. This ratio is nearly the best possible in thiscase, as we present a lower bound of

$\frac{3\sqrt{3}}{2} \approx 2.598$on the competitive ratio of any deterministic algorithm for thisproblem. We remark that the competitive ratio that we obtain in thiscase is much better than in the case of arbitrary number of sites, wherethe best possible competitive ratio is 4.

The algorithm is based on discounting future benefits. It is similar tothe future discounting algorithm presented above (specialized to thecase L=1) with the exception that the discount factor is taken to be φ.The analysis for this case adds some complication to that of the generalcase, in order to take advantage of the limited number of sites. Forexample, a discount factor of 2 optimizes the analysis of the sectionrelating to the future discounting algorithm, and then the competitiveratio is shown to be at most 4. The actual competitive ratio in thiscase is 3, which can be shown by an analysis similar to the one that wegive here.

The following theorem states the improved competitive ratio for thiscase.

Theorem 4. The discounting algorithm with discount factor φ achievescompetitive ratio r=1+φ˜2.618 for the web server farm problem with oneserver and two sites.

Proof. Let us first describe the algorithm in more detail. We associatewith each possible allocation its anticipated benefit, which combinesthe immediate benefit of this allocation and its future benefit. Tocompensate for the uncertainty of a future benefit (the algorithm mightlater decide to not collect these benefits), we discount it (withrespect to an immediate benefit) by a factor of φ. In other words, anaction that causes the algorithm to collect a benefit b and to have thepotential to collect in the future a benefit b′ has anticipated benefitof φ·b+b′.

Suppose that the situation of the online algorithm at some time t is asdescribed below. Assume, without loss of generality, the allocation ofthe server in the previous time unit is to the first site, as denoted bya triangle. Since the lookahead is L=1, the algorithm knows only thedemands in the current time and in the next t+1.

$d_{i,t} = \begin{bmatrix}\ldots & {\vartriangleright x} & x^{\prime} & ? & ? & \ldots \\\ldots & y & y^{\prime} & ? & ? & \ldots\end{bmatrix}$The algorithm decides on the allocation of the server at time t asfollows. If φ·x+x′≧y, then the previous allocation is kept, i.e., theonline algorithm allocates the server at time t to the first site.Otherwise the algorithm changes the allocation and the server isallocated to the second site. Observe that this decision matches thelargest anticipated benefit for the algorithm. If it decides to staywith the allocation as before, it collects a benefit of x and will havethe option to collect at next time t+1 an additional benefit x′,yielding an anticipated benefit φ·x+x′. If the algorithm decides tochange the allocation to the other site, then it collects no immediatebenefit (at this point there is no way for the online algorithm tocollect the benefit y), and in the next time t+1 it will have the optionto collect a benefit y′, yielding an anticipated benefit φ·0+y′=y′.

We will prove by induction on T that

${{r{\sum\limits_{t \leq T}{ON}_{t}}} + {q_{T + 1} \cdot {ON}_{T + 1}^{*}}} \geq {{\sum\limits_{t \leq T}{OFF}_{t}} + {OFF}_{T + 1}^{*}}$where ON_(t) is the actual benefit collected at time t by the onlinealgorithm; ON_(t) ^(*) is the potential benefit that the onlinealgorithm will collect at time t if it remains allocated to the samesite as in time t−1; OFF_(t), OFF_(t) ^(*) are defined similarly for anarbitrary offline algorithm; and q_(t)=φ if at time t−1 the online andthe offline solutions have the same allocation (i.e., if they have thesame “starting position” for time t), and q_(t)=1 otherwise. (Note thatON_(t) is either ON_(t) ^(*) in case the potential benefit is gained or0.) In a sense, this is a potential function analysis; see for exampleA. Borodin and R. El-Yaniv, On-line Computation and CompetitiveAnalysis, Cambridge University Press (1998).

Proving Equation (4) would complete the proof, since we can assume thatthe demands matrix is padded with zeros from both sides, and obtain forthe last time T that

${{r{\sum\limits_{t}{ON}_{t}}} \geq {\sum\limits_{i}{OFF}_{t}}},$as required.

The base case of the induction follows from padding the demands matrixwith zeros from both sides. For the inductive step, i.e., that theinduction hypothesis (4) for time T follows from that of time T−1, itsuffices to show thatr·ON _(T) +q _(T+1) ·ON _(T+1) ^(*) −q _(T) ·ON _(T) ^(*) ≧OFF _(T) +OFF_(T+1) ^(*) −OFF _(t) ^(*)  (5)We prove inequality (5) by verifying it over all cases. Denote thebenefits at times T, T+1 by x, x′, y, y′ as above. Assume without lossof generality that at time T−1 the server is allocated to the firstsite. The total number of cases is eight since there are two possibledecisions for the online, and four possible decisions for the offline.

Consider first the cases in which the allocation of the offline at timesT and T+1 are the same. Then at time T the offline collects the benefitOFF_(t) ^(*), and hence OFF_(t)=OFF_(t) ^(*). In addition ON_(t) ^(*)=x,so inequality (5) simplifies tor·tON _(T) +q _(T+1) ON _(T+1) ^(*) −q _(T) ·x ≧OFF _(T+1) ^(*)

-   1.q_(T)=φ, q_(T+1)=φ. Then both offline and online collect x. We    need to show that    r·x+φ·x′−φ·x≧x′-   i.e., that x+(φ−1)x′>0 which clearly holds. ✓-   2. q_(T)=1, q_(T+1)=1. Then offline collects y and online    collects x. We need to show that    r·x+x′−x≧y′-   i.e., that φ·x′≧y′, which holds since online keeps the previous    allocation. ✓-   3. q_(T)=φ, q_(T+1)=1. Then offline collects x and online does not    collect x. We need to show that    r·0+y′−φ·x≧x′-   which holds since online changes allocation. ✓-   4. q_(T)=1, q_(T+1)=φ. Then offline collects y and online does not    collect x. We need to show that    r·0+φ·y′−x≧y′-   i.e., that y′≧1/(φ−1)·x=φ·x which holds since online changes    allocation. ✓

Consider now the cases in which the allocation of the offline at times Tand T+1 are different. Then the benefit of the offline at time T isOFF_(T)=0. In addition ON_(T) ^(*)=x, so inequality (5) simplifies tor·ON _(T) q _(T+1) ON _(T+1) ^(*) −q _(T) ·x≧OFF _(T+1) ^(*) −OFF _(t)^(*)

-   5. q_(T)=φ, q_(T+1)=φ. Then offline and online both do not    collect x. We need to show that    r·0+φ·y′−φ·x≧y′−x-   i.e., that y′≧x, which clearly holds since online changes    allocation. ✓-   6. q_(T)=1, q_(T+1)=1. Then offline does not collect y and online    does not collect x. We need to show that    r·0+y′−x≧x′−y-   and since online changes allocation we indeed get y′≧φx+x′≧x+x′−y. ✓-   7. q_(T)=φ, q_(T+1)=1. Then offline does not collect x and online    collects x. We need to show that    r·x+x′−φ·x≧y′−x-   i.e. that 2x+x′≧y′ which holds since online keeps the previous    allocation. ✓-   8. q_(T)=1, q_(T+1)=φ. Then offline does not collect y and online    collects x. We need to show that    r·x+φ·x′−x≧x′−y-   i.e. that φ·x+(φ−1)x′+y≧0 which clearly holds. ✓    Multiple Servers with Lookahead One

We present an online algorithm for the web server farm problem withmultiple servers and lookahead one.

For the case of an even k, we describe below the algorithm and itsanalysis. In the case of an odd k>3, the algorithm is more involved. Weremark that it applies the algorithm below (for an even number ofservers) on the “first” k−1 servers, and yet another algorithm for theadditional server.

For an even k, the algorithm splits the set of servers into two sets ofequal size. The first set of servers is allocated so as to alwayscollect the k/2 biggest benefits (demands) on the odd time steps. Theseservers spend the even time steps in reallocation to those sites wherethey will be able to collect in the following time step, which is odd,the k/2 biggest benefits. On even time steps, these servers may collectzero benefit in the worst case. The second set of servers is scheduledin a symmetric way. It collects the k/2 biggest benefits on the eventime steps and spend the odd time steps on reallocation to the bestsites for the following time step which is even. We conclude that ateach time t, the total benefit that the k servers collect is at least aslarge as the k/2 biggest benefits at this time t, which is clearly atleast half the benefit that an optimal offline algorithm can collect atthis time t. Therefore, the competitive ratio of this algorithm is atmost 2.

Theorem 5. There is a deterministic algorithm for the web server farmproblem that achieves competitive ratio 2 when the number of servers kis even, and competitive ratio c_(k)=2+2/(2k−1) when k≧3 is odd.

Proof. The case of an even k was already shown above. We thus deal belowwith the case that k is odd.

For an odd k, the algorithm splits the servers into three groups. Twogroups are of size (k−1)/2 each, and the third group contains oneserver. The first group of servers collects the (k−1)/2 biggest benefitson the odd time steps. The second group collects the (k−1)/2 biggestbenefits on the even time steps. Let F_(t) denote the (k+1)/2 biggestbenefit on step t. The last server tries to collect F_(t) on each timestep, as follows. On the step t the server either collects F_(t) or itmoves to the web site q such that d_(q, t+1)=F_(t+1). The servercollects F_(t) if 2F_(t)>F_(t+1); otherwise, the server moves to the website q. If the server collects F_(t) on some step t, it wastes the stept+1 on a move to the (k+1)/2′th best web site q for the step t+2, i.e.,d_(q, t+2)=F_(t+2).

For the sake of analysis we split time into phases: the first phasestarts at the time step 0 (by convention we assume that benefits at thistime step are equal to zero), phase ends at the time step t+1 if theserver from the third group collects Ft on step t, the next phase startsat step t+2 and so on. Notice that the server from the third groupalways starts the phase allocated at the (k+1)/2′th best web site.Consider the first phase. We claim that

$F_{t} \geq {\frac{1}{4}{\sum\limits_{i = 1}^{t + 1}{F_{i}.}}}$Indeed, 2F_(i)≦F_(i+1) for i=1, . . . , t−1 and 2F_(t)>F_(t+1);therefore,

${{\sum\limits_{i = 1}^{t + 1}F_{i}} < {F_{t}{\sum\limits_{i = 1}^{t + 1}\frac{1}{2^{t - i}}}} \leq {2F_{t}{\sum\limits_{j = 0}^{\infty}\frac{1}{2^{j}}}}} = {4{F_{t}.}}$For p≦k, if S(τ, p) is a sum of p best benefits on step τ, then

${\sum\limits_{\tau}{S\left( {\tau,p} \right)}} \geq {\frac{p}{k}{\sum\limits_{\tau}{OFF}_{\tau}}}$and

${{\sum\limits_{\tau = 1}^{t + 1}{ON}_{\tau}} \geq {{\sum\limits_{\tau = 1}^{t + 1}{S\left( {\tau,\frac{\left( {k - 1} \right)}{2}} \right)}} + F_{t}} \geq {{\frac{3}{4}{\sum\limits_{\tau = 1}^{t + 1}{S\left( {\tau,\frac{\left( {k - 1} \right)}{2}} \right)}}} + {\frac{1}{4}{\sum\limits_{\tau = 1}^{t + 1}{S\left( {\tau,\frac{\left( {k + 1} \right)}{2}} \right)}}}} \geq {\left( {\frac{3\left( {k - 1} \right)}{8k} + \frac{\left( {k + 1} \right)}{8k}} \right){\sum\limits_{\tau = 1}^{t + 1}{OFF}_{\tau}}}} = {{\frac{{2k} - 1}{4k}{OFF}} = {c_{k}^{- 1}{{OFF}.}}}$The same proof works for any phase and therefore we proved that ouralgorithm is c_(k)-competitive for odd k≧3.

Randomized Online Algorithms

In this section we derive tight bounds on the competitive ratio ofrandomized online algorithms with lookahead L. These bounds are statedin the following theorem We remark that our randomized algorithmrequires only O(log L) random bits, independent of the input length.

Theorem 6. There is a randomized online algorithm with competitive ratio1+1/L. Furthermore, no randomized algorithm achieves a lower competitiveratio.

Proof. The idea behind the algorithm is that with lookahead L, we cancompute an optimal path for L time steps. However, we may have to make apoor move, in which we collect little or no benefit, in order to reachthe first state on this path. After following this path, we can againchoose the best state to move to during the next time step, possiblywithout collecting any benefit at all, but ensuring the best startingpoint for the next L steps. We lose only a 1/(L+1) fraction of theavailable benefit over each L+1 time steps. We refer to this as the“resetting” algorithm, since it operates in stages, after each of whichit resets to an optimal state for the upcoming stage. The onlinealgorithm will collect at least as much benefit as the optimal offlinealgorithm on every step except the resetting steps. Notice that thereare L+1 possible “phases”, i.e., phase j means (potentially) giving upall benefit during steps i·(L+1)+j for each i, where 0≦j≦L. Randomizingover these choices, the online algorithm loses in expectation only a1/(L+1) fraction of the offline benefit, and the competitive ratio is atmost

$\frac{1}{1 - \frac{1}{\left( {L + 1} \right)}} = {1 + {\frac{1}{L}.}}$

For any ∈>0, we demonstrate a lower bound of (L+1)/(L+∈) for theone-server problem. Our randomized adversary generates a demand matrixdi, t with ┌1/∈┐ rows and L+2 columns (which can be repeatedindefinitely), and a revenue matrix of all ones. The first column of thedemand matrix contains all zeroes, the next L columns contain all ones,and exactly one randomly chosen row in the last column contains a oneand the rest contain zeroes. The optimal choice is for the server tomove during the first step to the row containing the one in column L+2,and to stay in that row for L+1 steps, collecting a total benefit ofL+1. Any randomized online strategy stands only an ∈ chance of choosingthis row, and with probability at least 1−∈ must either miss the benefitin this column, or forgo benefit in some earlier column in order tocollect it. Thus, the expected benefit collected by the randomizedonline algorithm is at most L+∈.

Returning now to the drawings, the computation of new allocations(function block 23 in FIG. 2) is shown in FIG. 3. The process beginswith an initialization step in function block 31 where t is set to thecurrent time. In function block 32, the discounted revenues D(c, t+i)are computed for each customer c and each time period from current timet through current time plus lookahead t+L, using actual revenues R(c,t+i) and discount factor alpha:

${D\left( {c,{t + i}} \right)} = \frac{R\left( {c,{t + i}} \right)}{\alpha}$Based on this computation, the optimal allocations from the current timethrough current time plus lookahead are found, starting from the currentallocation, in function block 33. Finally, in function block 34,allocations and transitions in allocations for the next time period fromt to t+1 are returned.

While the invention has been described in terms of a preferredembodiment, those skilled in the art will recognize that the inventioncan be practiced with modification within the scope of the appendedclaims.

1. A computer implemented method of resource allocation to yield abenefit comprising the steps of: associating each customer's demand witha benefit gained; finding a time-varying allocation of resources thatwould yield a benefit which is based on the benefit gained associatedwith one or more customer's demands; implementing the time-varyingallocation of resources amongst one or more customers to yield saidbenefit; discounting future benefits; and finding optimal allocations ofresources from current time through current time plus lookahead based ondiscounted benefit and forecast demand, wherein the step of discountingfuture benefits is based on a future discounting algorithm, wherein thefuture discounting algorithm is a deterministic algorithm that achievesa competitive ratio of (1+1/L)(L+1)^(1/L), where L is a lookahead factorwhich models some amount of future demand known to a provider of theresource.
 2. A computer implemented method of resource allocation toyield a benefit comprising the steps of: associating each customer'sdemand with a benefit gained; finding a time-varying allocation ofresources that would yield a benefit which is based on the benefitgained associated with one or more customer's demands; implementing thetime-varying allocation of resources amongst one or more customers toyield said benefit; discounting future benefits; and finding optimalallocations of resources from current time through current time pluslookahead based on discounted benefit and forecast demand, wherein thestep of discounting future benefits is based on a future discountingalgorithm, wherein the algorithm is an intermittent reset algorithm thatachieves a competitive ratio of 1+4/(L−7), where L is a lookahead factorwhich models some amount of future demand known to a provider of theresource.
 3. A computer implemented method of resource allocation toyield a benefit comprising the steps of: modeling a resource allocationproblem mathematically; in the model obtained from said modeling step,dividing time into intervals of fixed length based on the assumptionthat demand is uniformly spread throughout each such interval; andassociating each customer's demand with a benefit gained; and finding atime-varying allocation of resources that would maximize a benefit whichis based on the benefit gained associated with one or more customer'sdemands; implementing the time-varying allocation of resources amongstone or more customers to maximize said benefit; discounting futurebenefits; and finding optimal allocations of resources from current timethrough current time plus lookahead based on discounted benefit andforecast demand, wherein the step of discounting future benefits isbased on a future discounting algorithm, wherein the future discountingalgorithm is a deterministic algorithm that achieves a competitive ratioof (1+1/L)(L+1)^(1/L), where L is a lookahead factor which models someamount of future demand known to a provider of the resource.
 4. Acomputer implemented method of resource allocation to yield a benefitcomprising the steps of: modeling a resource allocation problemmathematically; in the model obtained from said modeling step, dividingtime into intervals of fixed length based on the assumption that demandis uniformly spread throughout each such interval; and associating eachcustomer's demand with a benefit gained; and finding a time-varyingallocation of resources that would maximize a benefit which is based onthe benefit gained associated with one or more customer's demands; andimplementing the time-varying allocation of resources amongst one ormore customers to maximize said benefit, wherein the algorithm is anintermittent reset algorithm that achieves a competitive ratio of1+4/(L−7), where L is a lookahead factor which models some amount offuture demand known to a provider of the resource.
 5. A method forserver allocation in a Web server “farm” based on limited informationregarding future loads to achieve close to greatest possible revenuebased on an assumption that revenue is proportional to the utilizationof servers and differentiated by customer class comprising the steps of:modeling the server allocation problem mathematically; in the model,dividing time into intervals of fixed length based on the assumptionthat each site's demand is uniformly spread throughout each suchinterval; maintaining server allocations fixed for the duration of aninterval, servers being reallocated only at the beginning of aninterval, and a reallocated server being unavailable for the length ofthe interval during which it is reallocated providing time to “scrub”the old site (customer data) to which the server was allocated, toreboot the server and to load the new site to which the server has beenallocated, each server having a rate of requests it can server in a timeinterval and customers share servers only in the sense of using the sameservers at different times, but do not use the same servers at the sametime; and associating each customer's demand with a benefit gained bythe service provider in case a unit demand is satisfied and finding atime-varying server allocation that would maximize benefit gained bysatisfying sites' demand, wherein the future discounting algorithm is adeterministic algorithm that achieves a competitive ratio of(1+1/L)(L−1)^(1/L), where L is a lookahead factor which models someamount of future demand known to a provider of the resource.
 6. A methodfor server allocation in a Web server “farm” based on limitedinformation regarding future loads to achieve close to greatest possiblerevenue based on an assumption that revenue is proportional to theutilization of servers and differentiated by customer class comprisingthe steps of: modeling the server allocation problem mathematically; inthe model, dividing time into intervals of fixed length based on theassumption that each site's demand is uniformly spread throughout eachsuch interval; maintaining server allocations fixed for the duration ofan interval, servers being reallocated only at the beginning of aninterval, and a reallocated server being unavailable for the length ofthe interval during which it is reallocated providing time to “scrub”the old site (customer data) to which the server was allocated, toreboot the server and to load the new site to which the server has beenallocated, each server having a rate of requests it can server in a timeinterval and customers share servers only in the sense of using the sameservers at different times, but do not use the same servers at the sametime; and associating each customer's demand with a benefit gained bythe service provider in case a unit demand is satisfied and finding atime-varying server allocation that would maximize benefit gained bysatisfying sites' demand, wherein the algorithm is an intermittent resetalgorithm that achieves a competitive ratio of 1+4/(L−7), where L is alookahead factor which models some amount of future demand known to aprovider of the resource.